In 1948, F. John [1]} proved that any convex body in \({\mathbb {R}}^n\) 
contains a unique ellipsoid of maximum volume. Also he gave characterization of those convex bodies for which a maximal ellipsoid is the unit Euclidean ball
\(B_n\)  (in details see, e. g.,[2]}, [3]}). John's theorem implies the analogous statement which characterizes a unique minimum volume ellipsoid containing a given convex body.
